Model of a spherical universe

Curved Universe Model*

No one, knows whether the universe is flat or just has a very very large radius of curvature….. No one! Clichés held by many who believe that the universe is flat, such as being able to theoretically see the back of your head through a telescope in a positively curved universe, have made it hard to seriously consider such a scenario. As you will discover, such clichés could be somewhat misguided and indeed would not be possible under the framework of the spherically curved universe model that is proposed here.

Consider the following thought experiment…...

A flat universe, ruling out for the moment torus shapes and the like, stretches out infinitely. Based on the Cosmological Principal of homogeneity, such a flat universe would contain an infinite number of stars in an infinite number of galaxies. Conversely, a positively curved universe would host a finite number of stars and galaxies.

So, could a spherically curved finite universe be perfectly homogeneous? If you were to separate such a universe into 2 halves and measure the mass of each half, would they be identical? It would probably be safe to say not very likely. And if not equal, would it be possible for the universe to have a significant concentration of mass in one of the 'halves'…. By significant we mean a black hole with the mass equivalent of a million billion billion suns. We know the early universe was very very smooth, which would imply that such an asymmetry could not have existed. So how could such large mass located outside our observable universe be even remotely consistent with the many of the observations and discoveries made to date. It turns out that it may not only be consistent with but perhaps even help to explain some of the observations for which to date there has bee no satisfactory explanation. But first a bit of background.

No one, knows whether the universe is flat or just has a very very large radius of curvature….. No one! Clichés held by many who believe that the universe is flat, such as being able to theoretically see the back of your head through a telescope in a positively curved universe, have made it hard to seriously consider such a scenario. As you will discover, such clichés could be somewhat misguided and indeed would not be possible under the framework of the spherically curved universe model that is proposed here.

Consider the following thought experiment…...

A flat universe, ruling out for the moment torus shapes and the like, stretches out infinitely. Based on the Cosmological Principal of homogeneity, such a flat universe would contain an infinite number of stars in an infinite number of galaxies. Conversely, a positively curved universe would host a finite number of stars and galaxies.

So, could a spherically curved finite universe be perfectly homogeneous? If you were to separate such a universe into 2 halves and measure the mass of each half, would they be identical? It would probably be safe to say not very likely. And if not equal, would it be possible for the universe to have a significant concentration of mass in one of the 'halves'…. By significant we mean a black hole with the mass equivalent of a million billion billion suns. We know the early universe was very very smooth, which would imply that such an asymmetry could not have existed. So how could such large mass located outside our observable universe be even remotely consistent with the many of the observations and discoveries made to date. It turns out that it may not only be consistent with but perhaps even help to explain some of the observations for which to date there has bee no satisfactory explanation. But first a bit of background.

"Einstein's Ring" - Einstein predicted that in the presence of matter or mass, space is curved. The photograph [left] of LRG 3-757, taken by the Hubble telescope, provides visual evidence of the curving of space by the distortion of light rays from a distant blue galaxy being bent by the gravitational mass of another red galaxy located between us an the more distant one now forming what appears as a 'ring' shape.

**The Framework**

There are 3 basic alternatives for the universe's shape. Given that we are looking at alternatives to the flat model, leaves only 2 to consider, negatively and positively curved space. Because, as you will see, the anomalies and subsequent arguments seem to favour more

**the spherical positively curved model**, the model to the far left will be the focus of our attention in this thought experiment. This model will be termed the 4-D model as the 'curve' is occurring in a dimension over and above the 3 we are familiar with.

One of the main differences between a truly flat model and a positively curved model is the open and closed nature, respectively. The positively curved model represents a finite universe with a finite amount of mass/energy. If this mass/energy is not distributed perfectly evenly across the surface of the model, it would result in a hemispheric bias in terms of density. In other words, consider the positively curved model constructed with a perfectly spherical shape but with the surface having its mass distributed in nodules, with the nodules of mass not quite evenly distributed. Placed on a table it would stop rolling in a position with its 'centre' of mass directly above the point touching the table. For discussion purposes, we shall label this point at the 'bottom' touching the table, the 'south' pole.

A model based on a positively curved universe with a homogeneous distribution of matter, or an even moderately asymmetrical distribution of matter would not provide a viable framework. It would be at odds with a significant number of observations and as such is not a contender to the flat model. So why then even consider a positively curved model?

The goal is to establish a model that combines the flat nature of our observations with the finite nature of a positively curved model. But these characteristics of flat and curved run contradictory to each other. Or so it would seem.

So consider for a moment a universe that is positively curved and significantly inhomogeneous, carrying about two thirds of its mass at the south pole in the form of a universe-sized black hole. If this was the case, though, it would appear that we would first have to throw out almost all of the building blocks for the standard model in physics. Not a great starting point. The big bang, we are almost certain and for all intents and purposes, had to be perfectly symmetrical as observed from our location. One massive nodule on the side (or bottom) of our model is simply a non-starter. Even at 380,000 years we now know that the universe was still as smooth as a billiard ball.... In fact, much much smoother.

**Big Bang Shock Wave**

But what if instead of having a bump on one side, the big bang generated a perfectly symmetrical perfectly spherical

**shock wave**that propagated outward. Let's assume that this compression wave was perfectly formed and infinitely thin, but still carrying two thirds of the mass of the universe; the surface being similar to a black hole but instead of being a point, it is a spherical surface, like a soap bubble that is growing. This expanding shell will be labeled the 'black surface'. There are a couple of problems with this, however. Firstly, the shock wave has nothing in which to propagate out into. Secondly, a shock wave of this description, ignoring the fact that there is no space for it to propagate into, would at first glance be impossibly unstable. Certainly in a flat universe such a structure would be unstable. But do the the apparent constraints of instability and wave propagation medium still apply in a positively curved universe?

Before considering the two questions of how a shock wave can propagate if there is no space into which to propagate and, even if it could somehow propagate, how impossibly unstable it would be, we need to look at how a supernova might appear in the flat model. We know that all supernovas have a surge of electromagnetic energy at their leading edge or wave front. We are most familiar with the energy 'shock wave' of a supernova that lies within the visible portion of the electromagnetic spectrum. A super nova's spherical 'halo' or volume of influence might have a diameter of several billion light years while its 'shock wave' in the visible spectra might only be about 1/20th of a light year in thickness (a ratio of shell thickness to diameter of 1:40,000,000,000). Similarly, the matter that gets ejected during a supernova event can also exhibit a density wave at the leading edge (as evidenced in the Hubble photograph to the left). Furher, the balance of material remaining within the halo appears to be much more randomly distributed, almost nondescript.

If you were to model the propagation of a supernova using our flat universe model it's shock wave might best be represented by a ring... expanding outward on the flat Ωo=1 model.

Similarly, if you were to model the propagation of a supernova explosion using a positively curved universe Ωo>1 model, the shock wave might look like a ring propagating outward from a point. (Each sphere shown here indicating successive points in time, moving from left to right.) If the supernova were to occur at the 'north pole' of this model, we would expect the shock wave to propagate southwards, towards the 'equator'

If the universe's expansion is factored in, ignoring for the moment, the relative rates of universe's expansion and the rate of propagation of the supernova explosion itself, the supernova's shock wave could still propegate into the hemisphere opposite to where it started from (i.e. move from the north pole into the southern hemisphere).

Like someone maintaining their position relative to the loading platform by walking backward on a train as it moves out of the station, you could engineer a model such that space expanded at a rate that would exactly offset that of the supernova's propagation. If just the right balance could be struck, the supernova's shock wave would remain at a constant 'distance' (d) above the south pole, a kind of 'standing wave' as space 'moves' by.

Now, instead of a supernova, consider this shock wave in the context of the big bang. The distance 'd' would have to decrease. In fact, with both the expansion of space and big bang shock wave commencing at time zero, the distance 'd' would have to approach zero. Under this scenario the black surface is really just a standing wave, at a point in this 4-D model. Employing this model allows the big bang shock wave to 'propagate' relative to other points in space, simply with the expansion of the model itself. In other words, the black surface would be pushing the 'boundary' literally defining space as it grows. And secondly, because the black surface is at the same time a point in a positively curved universe, it is inherently stable.

**The 4 Dimensional Model and Our Location**

Up until recently, this proposed model would have had to be ruled out on the grounds that it predicts both an inhomogeneous and anisotropic universe. Most observations have strongly supported a homogeneous and isotropic universe. If more detailed observations continue to indicate a homogeneous and isotropic, the only way this model could possibly be correct would for us to be located exactly at the 'north' pole (centre of the 3-D model below), a position 'furthest' from the south/black pole. The odds of this occurring would be so infinitesimally small as to make it not worth considering. However, with growing evidence potentially indicating that our observable universe may indeed be slightly lopsided, would mean we no longer have to occupy such a prohibitively privileged position in the cosmos.

You often hear scientists describe a positively curved universe as being one that would allow you to see the back of your head through a telescope. This would not be possible under this model and is misleading, for two reasons.

Firstly, observations made in any direction from any point in this universe would be bent to travel along light paths that would end up passing into the blackpole (red dot at the south pole in the diagram). The closer an observer's position in the universe to the black pole, the tighter the radius of curvature of the light paths. Observations in the 'north' and 'south' directions would travel along a great circle coincident with the two poles while observations in the 'east' and 'west' directions would have the greatest degree of 'bending'.

Secondly, our view is limited to our observable portion of the universe, which would be represented by a disc on the surface of this model centred on our position in the Milky Way.
One of the interesting aspects of this model is that, with two thirds of the mass tied up in the black surface (black pole), there does not appear to be any requirement for dark energy. All matter would accelerate towards the black pole under the Newtonian model. From the perspective of any observer, however, the expansion would appear outward. It is also interesting that using WMAP's average value for the density parameter Ωo of 1.02 , yields a radius for the universe of 133 bly. The event horizon that would be associated with such a black pole having a mass of 2/3 rds that of the universe just happens to match the distance from the black pole to the edge of the observable universe. The curved surface that the model depicts is not a curve in the 3 dimensions we are familiar with. Rather it is a curve in a 4th dimension. The additional dimension provides the framework that allows our familiar 3 dimensional space to curve back in on itself (refer to 4-D animation below). Our limited 3 dimensional minds have great difficulty appreciating a 4 dimensional model but we are not limited to 3 dimensions from a mathematical perspective. The determination of the radius of the universe is shown under the 'Gravitational Thoughts" menu item on the page Curvature Probability. |

**The 3 Dimensional Model**

The corresponding 3-dimensional model is shown below. Note that when this 3-D model is combined with the inherent characteristics a positively curved universe, traveling in a 'straight' line outward from our current position, (near but not at the centre of the 3-D model) when you reach the black surface, you would be approximately half way to getting back to where you started from. No matter which direction you choose this would hold true. In this context the black surface represents the halfway

__point__, which is labled the black pole in our 4-D model above.

In the model below the center corresponds to the north pole and the outermost black surface, the south pole.

**Wrapping of Space**

An observer positioned exactly at the 'north' pole of the 4-D model (or centre of the 3-D model), would not be able to detect the black pole / black surface no matter how sophisticated the observer's equipment happened to be. From all other positions in the universe you would also not be able to detect the black pole directly. Asymmetries would simply increase as you moved off of the 'north' pole (i.e. away from the centre). Off centre, there would be hemispheric anomalies in both overall temperature and maturity of fluctuations in the Planck data. Observations in all directions would indicate an accelerating expanding universe with increased red-shifting at greater distances (towards the edge of the observable universe and towards the black pole's/black surface's event horizon). Only observers whose position was close enough such that their observable portion of the universe included the black pole/black surface would see the distortion caused by it.

As you moved closer to the black pole/black surface, like approaching any black holes event horizon, the universe would appear to turn inside out.

The '4-d' animation shown above provides a slightly different cartoon depicting the wrapping of space back in on itself.

**Accelerating, Slowing, Accelerating Expansion**

There is another interesting feature tied to the Big Bang Shock Wave model. As space expanded and the distance between the black pole and the balance of matter increased, the gravitational forces between the two decreased. So during this period, the rate at which galaxies moved away from us decreased. More recent measurements of increasing accelerations may mean that the extent of space has passed its zenith and is on its decent towards the big crunch.

The relationship between the models size above (how large space is) and the corresponding acceleration rates throughout the last 13.82 billion years: high rates followed by decreased rates followed by increasing rates not only seems plausible under this model, but might offer an explanation of the seemingly varying value of "dark energy".

* Although it raises a number of questions, the model described on this web page, which proposes that all matter is simply accelerating under a Newtonian framework in a positively curved universe, is worth exploring further.

To that end feedback is not only welcome, its appreciated.